In 1963, Sydney Newman becomes the BBC's new Head of Drama. To plug a gap between the broadcasts of Grandstand and Juke Box Jury, he devises a new science-fiction series, to be entitled Doctor Who, which will cater to children and adults alike, and feature a central character who is an old man and a "doctor". Newman recruits Verity Lambert to head the programme as producer, despite the difficulties she faces from others in her new role. Lambert and Waris Hussein, the programme's director, recruit William Hartnell for the role of The Doctor, despite some trepidation from him over how it will aid his acting career.
After Lambert strengthens her assertiveness and secures a set piece for the programme that resembles the interior of the TARDIS. Production of the pilot episode An Unearthly Child is beset by difficulties. Newman dislikes the end results and orders a re-shoot, including a request for Hartnell's character to be gentler and kinder on screen. Lambert and Hussein manage to complete the re-shoot in time for the pilot to be broadcast on its scheduled transmission date. Upon learning that the programme is to be cancelled, due to the diminished audience the pilot episode received in the wake of the assassination of John F. Kennedy, Lambert successfully appeals to Newman for it to be broadcast again before the second episode is transmitted. Although Newman reluctantly goes against his belief not to include monsters into Doctor Who, by agreeing to let Lambert introduce creatures called the Daleks into the next serial, he admits he was mistaken upon seeing the strong viewing figures.
Meanwhile, Hartnell takes delight in his role, becoming attached to his character and the popularity it brings him with children. However, as the programme progresses, and both cast and crew change over time, Hartnell's health begins to decline. Concerned over his failure to properly remember his lines, Newman agrees to his superior's request to replace Hartnell for a new actor, Patrick Troughton (Reece Shearsmith), for the next series. With a heavy heart, Hartnell agrees to leave, but emotionally breaks down upon telling his wife the news. As Hartnell prepares for his final scene in 1966, recalling how Doctor Who began and his involvement with it, he commends Troughton for being his successor before his first scene is filmed. As filming begins, Hartnell looks to the side and sees a brief vision of Matt Smith, who will play the same role nearly 50 years later, silently acknowledge his work.
The film ends on an epilogue narrative of each of the main real-life figures in the story, before closing on the real Hartnell's speech made at the end of the serial The Dalek Invasion of Earth.